Umbral calculus is a basic technique in combinatorics, systematized by Gian-Carlo Rota [1]}. Generally speaking, for a given sequence \((A_n)_{n\in \mathbb {N}}\) , it asks us to treat \(A_n\)  as \(A^n\)  to simplify some combinatorics deduction and revealed some imperceptible results. [2]} gives an appropriate introduction about the applications, and [3]} gives a comprehensive survey. Consider Bernoulli numbers \((\mathrm {B}_n)_{n\in \mathbb {N}}\) , we can write Faulhaber's formula as
\(\sum _{k=1}^{n} k^m=\frac{1}{m+1}\sum _{k=0}^{m}\binom{m+1}{k}\mathrm {B}_k n^{m+1-k}=\frac{(\mathrm {B}+n)^{m+1}-\mathrm {B}^{m+1}}{m+1}.\) 
